Linear programming vs dual problem
In the following lines, I will describe what the variables x1,x2 mean for linear programming and what the variables y1,y2 mean for the dal problem.
Linear programming
In the case of linear programming, we need to maximise the profit from female and male hats.
Objective Function:
max 50𝑥1+75𝑥2
Constraints:
5𝑥1+10𝑥2≤480
9𝑥1+8𝑥2≤480
𝑥1,𝑥2≥0
In linear programming, the variables x1 and x2 represent the number of products.
Dual problem
In this case, we should minimize the total cost of increasing the capacities of cutting and knitting to meet the demand. This is because we want to find the optimal allocation of resources that minimizes the additional cost incurred in expanding these capacities while satisfying the given production requirements. Thus, the objective function in the dual problem is formulated to minimize the total cost associated with increasing the capacities, ensuring cost-effectiveness in resource allocation.
Objective Function:
min 480𝑦1+480𝑦2
Constraints:
5𝑦1+9𝑦2≥50
10𝑦1+8𝑦2≥75
𝑦1,𝑦2≥0
The y variables in the dual problem represent the shadow prices or the dual prices associated with the constraints of the primal problem. Specifically, y1 corresponds to the shadow price for the constraint on cutting capacity, while y2 corresponds to the shadow price for the constraint on knitting capacity. These shadow prices indicate the change in the objective function value of the primal problem per unit increase in the corresponding constraint's capacity, highlighting the economic value of additional resources in cutting and knitting.